Continuous Probability Functions and the Uniform Distribution

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Description

Once you have grasped the notion of a discrete random variable, it should be clear that all random variables need not be discrete. For example, consider measuring the atmospheric temperature at some prescribed location. The measured temperature would be random and could take on a continuum of values (theoretically speaking). Under these circumstances, we say that the random variable is continuous. All the machinery developed for discrete random values (such as expected value, variance, and mean) must be elevated to continuous random variables to handle this situation. The uniform distribution (which you have programmed using the random module) is an example of a continuous probability distribution.

Continuous Probability Functions
The Uniform Distribution

Continuous Probability Functions

We begin by defining a continuous probability density function. We use the function notation f(x). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function f(x) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, PROBABILITY = AREA.

Example 5.1

Consider the function $f(x) = \dfrac{1}{20}$ for 0 ≤ x ≤ 20. x = a real number. The graph of $f(x) = \dfrac{1}{20}$ is a horizontal line. However, since 0 ≤ x ≤ 20, f(x) is restricted to the portion between x = 0 and x = 20, inclusive.

Figure 5.5

$f(x) = \dfrac{1}{20}$ for 0 ≤ x ≤ 20.

The graph of $f(x) = \dfrac{1}{20}$ is a horizontal line segment when 0 ≤ x ≤ 20.

The area between $f(x) = \dfrac{1}{20}$ where 0 ≤ x ≤ 20 and the x-axis is the area of a rectangle with base = 20 and height = $\dfrac{1}{20}$ .

$AREA = 20(\dfrac{1}{20})=1$

Suppose we want to find the area between $f(x) = \dfrac{1}{20}$ and the x-axis where 0 < x < 2.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

Figure 5.6

$AREA = (2 – 0)(\dfrac{1}{20}) = 0.1$

$(2–0)=2=\text{base of a rectangle}$

Reminder

$\text{area of a rectangle = (base)(height)}$

The area corresponds to a probability. The probability that x is between zero and two is 0.1, which can be written mathematically as P(0 < x < 2) = P(x < 2) = 0.1.

Suppose we want to find the area between $f(x) = \dfrac{1}{20}$ and the x-axis where 4 < x < 15.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

Figure 5.7

$\text{AREA = }(15 – 4)(\dfrac{1}{20}) = 0.55$

(15 – 4) = 11 = the base of a rectangle

The area corresponds to the probability P(4 < x < 15) = 0.55.

Suppose we want to find P(x = 15). On an x-y graph, x = 15 is a vertical line. A vertical line has no width (or zero width). Therefore, $\text{ P(x = 15) = (base)(height) }= (0)(\dfrac{1}{20}) = 0$

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

Figure 5.8

P(X <= x), which can also be written as P(X < x) for continuous distributions, is called the cumulative distribution function or CDF. Notice the less than or equal to symbol. We can also use the CDF to calculate P(X > x). The CDF gives area to the left and P(X > x) gives area to the right. We calculate P(X > x) for continuous distributions as follows: P(X > x) = 1 – P (X < x).

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

Figure 5.9

Label the graph with f(x) and x. Scale the x and y axes with the maximum x and y values. $f(x) = \dfrac{1}{20}$ , $0 ≤ x ≤ 20$ .

To calculate the probability that x is between two values, look at the following graph. Shade the region between x = 2.3 and x = 12.7. Then calculate the shaded area of a rectangle.

This shows the graph of the function f(x) = 1/20. A horiztonal line ranges from the point (0, 1/20) to the point (20, 1/20).

Figure 5.10

$\text{ P (2.3 < x < 12.7) = (base)(height) }=(12.7−2.3)(\dfrac{1}{20})=0.52$

Try It 5.1

Consider the function $f(x) = \dfrac{1}{8}$ for $0 ≤ x ≤ 8$ . Draw the graph of f(x) and find P(2.5 < x < 7.5).

Source: OpenStax, https://openstax.org/books/statistics/pages/5-introduction
This work is licensed under a Creative Commons Attribution 4.0 License.

The Uniform Distribution

The uniform distribution is a continuous probability distribution and is concerned with events that are equally likely to occur. When working out problems that have a uniform distribution, be careful to note if the data are inclusive or exclusive of endpoints.

Example 5.2

The data in Table 5.1 are 55 smiling times, in seconds, of an eight-week-old baby.

10.4	19.6	18.8	13.9	17.8	16.8	21.6	17.9	12.5	11.1	4.9
12.8	14.8	22.8	20.0	15.9	16.3	13.4	17.1	14.5	19.0	22.8
1.3	0.7	8.9	11.9	10.9	7.3	5.9	3.7	17.9	19.2	9.8
5.8	6.9	2.6	5.8	21.7	11.8	3.4	2.1	4.5	6.3	10.7
8.9	9.4	9.4	7.6	10.0	3.3	6.7	7.8	11.6	13.8	18.6

Table 5.1

The sample mean = 11.49 and the sample standard deviation = 6.23.

We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. This means that any smiling time from zero to and including 23 seconds is equally likely. The histogram that could be constructed from the sample is an empirical distribution that closely matches the theoretical uniform distribution.

Let X = length, in seconds, of an eight-week-old baby's smile.

The notation for the uniform distribution is

X ~ U(a, b) where a = the lowest value of x and b = the highest value of x.

The probability density function is

$f(x) = \dfrac{1}{b−a}$ for a ≤ x ≤ b.

For this example, X ~ U(0, 23) and

$f(x) = \dfrac{1}{23−0}$ for 0 ≤ X ≤ 23.

Formulas for the theoretical mean and standard deviation are

$μ=\dfrac{a+b}{2}$ and

$σ=\sqrt{\dfrac{(b−a)^2}{12}}$

For this problem, the theoretical mean and standard deviation are

$μ =\dfrac{ 0 + 23}{2} = 11.50$ seconds and

$σ =\sqrt{\dfrac{(23 − 0)^2}{12}} = 6.64$ seconds.

Notice that the theoretical mean and standard deviation are close to the sample mean and standard deviation in this example.

Try It 5.2

The data that follow are the number of passengers on 35 different charter fishing boats. The sample mean = 7.9 and the sample standard deviation = 4.33. The data follow a uniform distribution where all values between and including zero and 14 are equally likely. State the values of a and b. Write the distribution in proper notation, and calculate the theoretical mean and standard deviation.

1	12	4	10	4	14	11
7	11	4	13	2	4	6
3	10	0	12	6	9	10
5	13	4	10	14	12	11
6	10	11	0	11	13	2

Table 5.2

Example 5.3

a. Refer to Example 5.2. What is the probability that a randomly chosen eight-week-old baby smiles between two and 18 seconds?

Solution 1

P(2 < x < 18) = (base)(height) =

$(18 – 2)(\dfrac{1}{23}) = \dfrac{16}{23}$

This graph shows a uniform distribution. The horizontal axis ranges from 0 to 15. The distribution is modeled by a rectangle

Figure 5.11

b. Find the 90^th percentile for an eight-week-old baby's smiling time.

Solution 2

b. Ninety percent of the smiling times fall below the 90^th percentile, k, so P(x < k) = 0.90.

$P(x < k)=0.90$

(base) (height) = 0.90

$(k−0)(\dfrac{1}{23})=0.90$

k=(23)(0.90)=20.7

This shows the graph of the function f(x) = 1/15. A horiztonal line ranges from the point (0, 1/15) to the point (15, 1/15).

Figure 5.12

c. Find the probability that a random eight-week-old baby smiles more than 12 seconds knowing that the baby smiles more than eight seconds.

Solution 3

c. This probability question is a conditional. You are asked to find the probability that an eight-week-old baby smiles more than 12 seconds when you already know the baby has smiled for more than eight seconds.

Find P(x > 12|x > 8) There are two ways to do the problem. For the first way, use the fact that this is a conditional and changes the sample space. The graph illustrates the new sample space. You already know the baby smiled more than eight seconds.

for 8 < x < 23

Write a new f(x):

$f(x) = \dfrac{1}{23 − 8} = \dfrac{1}{15}$
for 8 < x < 23.

$P(x > 12 | x > 8) = (23 − 12)(\dfrac{1}{15}) = \dfrac{11}{15}$

f(X)=1/15 graph displaying a boxed region consisting of a horizontal line extending to the right from point 1/15 on the y-axi

Figure 5.13

For the second way, use the conditional formula from Probability Topics with the original distribution.

$P(A | B) = \dfrac{P(A \text{ AND }B)}{P(B)}$

For this problem, A is (x > 12) and B is (x > 8).

So,

$P(x > 12 | x > 8) = \dfrac{ (x > 12 \text{ AND } x > 8 )}{P(x > 8)} =\dfrac {P(x > 12)}{P(x > 8)}=\dfrac{\dfrac{11}{23}} {\dfrac{15}{23}}=\dfrac{11}{15}$

This shows the graph of the function f(x) = 1/23. A horizontal line ranges from the point (0, 1/23) to the point (23, 1/23).

Figure 5.14

Try It 5.3

A distribution is given as X ~ U(0, 20). What is P(2 < x < 18)? Find the 90^th percentile.

Example 5.4

The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between zero and 15 minutes, inclusive.

a. What is the probability that a person waits fewer than 12.5 minutes?

Solution 1

a. Let X = the number of minutes a person must wait for a bus. a = 0 and b = 15. X ~ U(0, 15). Write the probability density function.

$f (x) = \dfrac{1}{15 − 0} = \dfrac{1}{15}$ for 0 ≤ x ≤ 15.

Find P (x < 12.5). Draw a graph.

$P(x < k)=\text{ (base)(height) }=(12.5−0)(\dfrac{1}{15})=0.8333$

The probability a person waits fewer than 12.5 minutes is 0.8333.

Figure 5.15

b. On the average, how long must a person wait? Find the mean, μ, and the standard deviation, σ.

Solution 2

$μ = \dfrac{a + b}{2} = \dfrac{15 + 0}{2} = 7.5$ . On the average, a person must wait 7.5 minutes.

$σ = \sqrt{\dfrac{(b−a)^2}{12}} = \sqrt{\dfrac{(15−0)^2}{12}} = 4.3$ . The standard deviation is 4.3 minutes.

c. Ninety percent of the time, the minutes a person must wait falls below what value?

This question asks for the 90^th percentile.

Solution 32

c. Find the 90^th percentile. Draw a graph. Let k = the 90^th percentile.

$P(x < k)=\text{ (base)(height) }=(k−0)(\dfrac{1}{15})$

$0.90=(k)(\dfrac{1}{15})$

$k=(0.90)(15)=13.5$

k is sometimes called a critical value.

The 90^th percentile is 13.5 minutes. Ninety percent of the time, a person must wait at most 13.5 minutes.

Figure 5.16

Try It 5.4

The total duration of baseball games in the major league in the 2011 season is uniformly distributed between 447 hours and 521 hours inclusive.

Find a and b and describe what they represent.
Write the distribution.
Find the mean and the standard deviation.
What is the probability that the duration of games for a team for the 2011 season is between 480 and 500 hours?
What is the 65th percentile for the duration of games for a team for the 2011 season?

Example 5.5

Suppose the time it takes a nine-year old to eat a donut is between 0.5 and 4 minutes, inclusive. Let X = the time, in minutes, it takes a nine-year-old child to eat a doughnut. Then X ~ U(0.5, 4).

a. The probability that a randomly selected nine-year-old child eats a doughnut in at least two minutes is _______.

Solution 1

a. 0.5714

b. Find the probability that a different nine-year-old child eats a doughnut in more than two minutes given that the child has already been eating the doughnut for more than 1.5 minutes.

The second question has a conditional probability. You are asked to find the probability that a nine-year-old child eats a doughnut in more than two minutes given that the child has already been eating the donut for more than 1.5 minutes. Solve the problem two different ways (see Example 5.3). You must reduce the sample space. First way: Since you know the child has already been eating the doughnut for more than 1.5 minutes, you are no longer starting at a = 0.5 minutes. Your starting point is 1.5 minutes.

Write a new f(x):

$f(x) = \dfrac{1}{4−1.5} = \dfrac{2}{5}$ for

$1.5 ≤ x≤4$ .

Find P(x > 2|x > 1.5). Draw a graph.

f(X)=2/5 graph displaying a boxed region consisting of a horizontal line extending to the right from point 2/5 on the y-axis,

Figure 5.17

P(x > 2 | x > 1.5) = (base)(new height) =

$(4 – 2)(\dfrac{2}{5})=\dfrac{4}{5}$

Solution 2

$\dfrac{4}{5}$

The probability that a nine-year-old child eats a donut in more than two minutes given that the child has already been eating the doughnut for more than 1.5 minutes is

$\dfrac{4}{5}$ .

Second way: Draw the original graph for X ~ U(0.5, 4). Use the conditional formula

$P(x>2 | x>1.5) = \dfrac{P(x>2 \text{ AND } x > 1.5)} {P(x > 1.5)} = \dfrac{P(x > 2)}{P(x > 1.5)}=\dfrac{\dfrac{2}{3.5}}{\dfrac{2.5}{3.5}}=0.8=\dfrac{4}{5}$

Try It 5.5

Suppose the time it takes a student to finish a quiz is uniformly distributed between six and 15 minutes, inclusive. Let X = the time, in minutes, it takes a student to finish a quiz. Then X ~ U(6, 15).

Find the probability that a randomly selected student needs at least eight minutes to complete the quiz. Then find the probability that a different student needs at least eight minutes to finish the quiz given that she has already taken more than seven minutes.

Example 5.6

Ace Heating and Air Conditioning Service finds that the amount of time a repairman needs to fix a furnace is uniformly distributed between 1.5 and four hours. Let x = the time needed to fix a furnace. Then x ~ U(1.5, 4).

Find the probability that a randomly selected furnace repair requires more than two hours.
Find the probability that a randomly selected furnace repair requires less than three hours.
Find the 30th percentile of furnace repair times.
The longest 25 percent of furnace repair times take at least how long? (In other words: find the minimum time for the longest 25 percent of repair times.) What percentile does this represent?
Find the mean and standard deviation

Solution 1

a. To find f(x):

$f (x) = \dfrac{1}{4 − 1.5} = \dfrac{1}{2.5}$ so

$f(x) = 0.4$

P(x > 2) = (base)(height) = (4 – 2)(0.4) = 0.8

Figure 5.18 Uniform distribution between 1.5 and four with shaded area between two and four representing the probability that the repair time x is greater than two

Solution 2

b. P(x < 3) = (base)(height) = (3 – 1.5)(0.4) = 0.6

The graph of the rectangle showing the entire distribution would remain the same. However the graph should be shaded between x = 1.5 and x = 3. Note that the shaded area starts at x = 1.5 rather than at x = 0. Because X ~ U(1.5, 4), x cannot be less than 1.5.

Figure 5.19 Uniform distribution between 1.5 and four with shaded area between 1.5 and three representing the probability that the repair time x is less than three

Solution 3

Figure 5.20 Uniform distribution between 1.5 and 4 with an area of 0.30 shaded to the left, representing the shortest 30 percent of repair times.

P (x < k) = 0.30
P(x < k) = (base)(height) = (k – 1.5)(0.4)
0.3 = (k – 1.5) (0.4); Solve to find k:
0.75 = k – 1.5, obtained by dividing both sides by 0.4
k = 2.25, obtained by adding 1.5 to both sides

The 30^th percentile of repair times is 2.25 hours. 30 percent of repair times are 2.5 hours or less.

Solution 4

A graph with an x and an f(x) axis is shown. A box is drawn on the graph between 1.5 and 4 on the x axis, and 0 and 0.4 on th

Figure 5.21 Uniform distribution between 1.5 and 4 with an area of 0.25 shaded to the right representing the longest 25 percent of repair times.

P(x > k) = 0.25
P(x > k) = (base)(height) = (4 – k)(0.4)
0.25 = (4 – k)(0.4); Solve for k:
0.625 = 4 − k,
obtained by dividing both sides by 0.4
−3.375 = −k,

obtained by subtracting four from both sides: k = 3.375

The longest 25 percent of furnace repairs take at least 3.375 hours (3.375 hours or longer).

Note: Since 25 percent of repair times are 3.375 hours or longer, that means that 75 percent of repair times are 3.375 hours or less. 3.375 hours is the 75^th percentile of furnace repair times.

Solution 5

$μ=\dfrac{a+b}{2}$ and

$σ=\sqrt{\dfrac{(b−a)^2}{12}}$

$μ\dfrac{1.5+4}{2} 2.75$ hours and

$σ=\sqrt{\dfrac{(4–1.5)^2}{12}}=0.7217$ hours

Try It 5.6

The amount of time a service technician needs to change the oil in a car is uniformly distributed between 11 and 21 minutes. Let X = the time needed to change the oil on a car.

Write the random variable X in words. X = __________________.
Write the distribution.
Graph the distribution.
Find P (x > 19).
Find the 50th percentile.

Continuous Probability Functions and the Uniform Distribution

Description

Table of contents

Continuous Probability Functions

Example 5.1

Reminder

Try It 5.1

The Uniform Distribution

Example 5.2

Try It 5.2

Example 5.3

Solution 1

Solution 2

Solution 3

Try It 5.3

Example 5.4

Solution 1

Solution 2

Solution 32

Try It 5.4

Example 5.5

Solution 1

Solution 2

Try It 5.5

Example 5.6

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Try It 5.6